All of the 4th grade teachers and students from Almond went on a field trip to an archaeology museum. Tickets were $$8.50$ each for teachers and $$4.50$ each for students, and the group paid $$70.50$ in total. The next month, the same group visited a natural history museum where the tickets cost $$25.50$ each for teachers and $$9.00$ each for students, and the group paid $$166.50$ in total. Find the number of teachers and students on the field trips.
Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${8.5x+4.5y = 70.5}$ ${25.5x+9y = 166.5}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-3$ ${-25.5x-13.5y = -211.5}$ ${25.5x+9y = 166.5}$ Add the top and bottom equations together. $ -4.5y = -45 $ $ y = \dfrac{-45}{-4.5}$ ${y = 10}$ Now that you know ${y = 10}$ , plug it back into $ {8.5x+4.5y = 70.5}$ to find $x$ ${8.5x + 4.5}{(10)}{= 70.5}$ $8.5x+45 = 70.5$ $8.5x = 25.5$ $x = \dfrac{25.5}{8.5}$ ${x = 3}$ You can also plug ${y = 10}$ into $ {25.5x+9y = 166.5}$ and get the same answer for $x$ ${25.5x + 9}{(10)}{= 166.5}$ ${x = 3}$ There were $3$ teachers and $10$ students on the field trips.